Repeated Eigenvalues

A repeated eigenvalue is simply one that shows up twice as a root of $\det(A - \lambda I) = 0$, like $(\lambda - 5)^2 = 0$. The trouble is that it often comes with only one eigenvector direction, and a $2\times2$ system needs two independent solutions. The fix, $\vec x_2 = t\,e^{\lambda t}\vec v + e^{\lambda t}\vec w$, looks heavy, but it's just two parts: $t\,e^{\lambda t}\vec v$ (the old straight-line solution, times $t$) and $e^{\lambda t}\vec w$ (a small correction). The vector $\vec w$ is called a generalized eigenvector because it's "almost" an eigenvector: $A - \lambda I$ doesn't send it to $\vec 0$, but it sends it to $\vec v$. Bear with it; the recipe is short.

Why the $t$? This parallels the $te^{rt}$ solution for repeated roots of a single equation, where $e^{rt}$ alone gave only one solution and multiplying by $t$ found the second. Here's why the correction $\vec w$ is needed too. Plug $\vec x_2$ into $\vec x' = A\vec x$. The left side is $e^{\lambda t}\vec v + \lambda te^{\lambda t}\vec v + \lambda e^{\lambda t}\vec w$, and the right side is $\lambda te^{\lambda t}\vec v + e^{\lambda t}A\vec w$. The $t$ terms match because $A\vec v = \lambda\vec v$. What's left says $\vec v + \lambda\vec w = A\vec w$, which is exactly $(A - \lambda I)\vec w = \vec v$.

A common mistake. It's tempting to try $te^{\lambda t}\vec v$ by itself as the second solution, because that's what worked for a single equation. It isn't enough: plugging it in leaves an extra $e^{\lambda t}\vec v$ that nothing cancels. That leftover is exactly what the $e^{\lambda t}\vec w$ term is there to absorb.

Pictures: with one eigenvector, the origin is an improper node: trajectories curve to become tangent to the single eigenvector line. If $A = \lambda I$, every vector is an eigenvector and trajectories are straight rays (a star).

Repeated eigenvalues mark critical damping in coupled systems: the borderline between oscillating and not, where controllers are often tuned.